Finite Group Databasebeta

← #14536838

Subgroups — #14536838

Order 2176.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587576C211yesmaximal, characteristic, complemented
2C2×#10587571C211yesmaximal, characteristic
3#11762744C211yesmaximal, characteristic, complemented
4#11762736C211yesmaximal, characteristic, complemented
5#11762734C211yesmaximal, characteristic, complemented
6#11762732C212yesmaximal, complemented
7#11762731C212yesmaximal, complemented
8#11762729C212yesmaximal, complemented
9#11762724C212yesmaximal
10#11762673C211yesmaximal, characteristic, complemented
11C17×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×#63227C2211yescharacteristic
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29C2×C4×Dic17C2211yescharacteristic
30#10587576—24no—
31#10587571—24no—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C2×C22⋊D17—22no—
44C2×C22⋊D17—42no—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—42no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17—42no—
54C22⋊C4×C17—22no—
55C22⋊C4×C17—42no—
56C22⋊Dic17—22no—
57C22⋊Dic17—42no—
58#63227—22no—
59#63227—22no—
60#63227—42no—
61#63227—42no—
62C4⋊Dic17—22no—
63C4⋊Dic17—24no—
64#63225—24no—
65#63225—42no—
66#63225—42no—
67C4×Dic17D414yes—
68C23×C17—21no—
69C23×C17—21no—
70C23×C17—22no—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17C2411yescharacteristic
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—21no—
77C22×D17—21no—
78C22×D17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17C2×D412yes—
89C22⋊D17—44no—
90C2×Dic17—22no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17C2×D412yes—
102C2×Dic17C2×D412yes—
103Dic34—44no—
104#2820—171nomaximal, Sylow
105C22×C17—41no—
106C22×C17—41no—
107C22×C17—41no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17C22×D411yescharacteristic
111C22×C17H(5,𝔽2)12yes—
112C22×C17H(5,𝔽2)11yescharacteristic
113C22×C172−1+411yescharacteristic
114C22×C172−1+412yes—
115C2×D17—41no—
116C2×D17—41no—
117C2×D17—41no—
118C2×D17—42no—
119C2×D17—42no—
120C4×C17—22no—
121C4×C17—41no—
122C4×C17—42no—
123C4×C17—42no—
124Dic17—22no—
125Dic17—22no—
126Dic17—41no—
127Dic17—42no—
128Dic17—42no—
129C2×#125—171no—
130C2×#124—171no—
131#399—171no—
132#399D1711yescharacteristic, complemented
133#397—171no—
134#397—171no—
135#396—172no—
136#396—172no—
137#395—172no—
138#394—172no—
139#382—171no—
140C2×C17—81no—
141C2×C17#53511yescharacteristic
142C2×C17#53611yescharacteristic
143C2×C17#53711yescharacteristic
144C2×C17#56212yes—
145C2×C17#56312yes—
146D17—81no—
147C22×Q8—171no—
148C22×D4—171no—
149#125—344no—
150#124—344no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—171no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4C2×D1711yescharacteristic
157C2×C4⋊C4C2×D1712yes—
158C2×C22⋊C4—171no—
159C2×C22⋊C4—171no—
160C2×C22⋊C4—171no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4C2×D1711yescharacteristic
165C2×C22⋊C4C2×D1712yes—
166C2×C42—171no—
167#95—172no—
168#95—171no—
169#95—172no—
170#95—172no—
171#95C2×D1711yescharacteristic
172C17#282011yescharacteristic, complemented, Sylow
173C24—171no—
174C24C22×D1711yescharacteristic
175C2×Q8—342no—
176C2×Q8—682no—
177C2×D4—342no—
178C2×D4—682no—
179C22×C4—171no—
180C22×C4—171no—
181C22×C4—172no—
182C22×C4—172no—
183C22×C4—171no—
184C22×C4C22×D1712yes—
185C22×C4C22×D1711yescharacteristic
186C22×C4C22×D1712yes—
187C22×C4C22×D1711yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—344no—
191C4⋊C4—344no—
192C4⋊C4—342no—
193C4⋊C4—682no—
194C4⋊C4—682no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—342no—
198C22⋊C4—342no—
199C22⋊C4—342no—
200C22⋊C4—682no—
201C22⋊C4—682no—
202C22⋊C4—682no—
203C42—174no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—341no—
210C23—342no—
211C23—341no—
212C23—341no—
213C23—342no—
214C23C23×D1711yescharacteristic
215Q8—684no—
216D4—684no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—172no—
227C2×C4—172no—
228C2×C4—342no—
229C2×C4—341no—
230C2×C4—341no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4D4×D1712yes—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—682no—
246C22—682no—
247C22—681no—
248C22—681no—
249C22—681no—
250C22C2×D4×D1711yescharacteristic
251C22#1058769112yes—
252C22#1058769411yescharacteristic
253C22#1058769611yescharacteristic
254C22#1058769712yes—
255C4—22no—
256C4—42no—
257C4—41no—
258C4—42no—
259C4—342no—
260C4—342no—
261C4—682no—
262C4—681no—
263C4—682no—
264C2—81no—
265C2—1361no—
266C2#1176337211yescharacteristic
267C2#1176338111yescharacteristic
268C2#1176338911yescharacteristic
269C2#1176355612yes—
270C2#1176356012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radicalC22×C22⋊D17C22
hypercenter#399D17
hypocenterC17#2820
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#10587576C2semidirect
3. #11762744C2semidirect
4. #11762736C2semidirect
5. #11762734C2semidirect
6. #11762732C2semidirect
7. #11762731C2semidirect
8. #11762729C2semidirect
10. #11762673C2semidirect
11. C17×#399C2semidirect
132. #399D17semidirect
172. C17#2820semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#105875762
C2×#105875712
#117627442
#117627362
#117627342
#117627322
#117627312
#117627292
#117627242
#117626732
C17×#3992
#282017